The Fibonacci recurrence f n = f n-1 + f n-2 has the associated matrix equation x
Question:
and
(a) With f0 = 0 and f1 = 1, use mathematical induction to prove that
for all n ¥ 1.
(b) Using part (a), prove that
fn+1fn-1 - f2n = (-1)n
for all n 1. [This is called Cassinis Identity, after the astronomer Giovanni Domenico Cassini (16251712). Cassini was born in Italy but, on the invitation of Louis XIV, moved in 1669 to France, where he became director of the Paris Observatory. He became a French citizen and adopted the French version of his name: Jean-Dominique Cassini. Mathematics was one of his many interests other than astronomy. Cassinis Identity was published in 1680 in a paper submitted to the Royal Academy of Sciences in Paris.]
(c) An 8 8 checkerboard can be dissected as shown in Figure 4.29(a) and the pieces reassembled to form the 5 13 rectangle in Figure 4.29(b).
The area of the square is 64 square units, but the rectangles area is 65 square units! Where did the extra square come from?
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